Method, device, and computer program for locating an emitting source of which measurements of emission propagation at locations different from that of the emitting source can be obtained from those locations, lacking space perception

ABSTRACT

A method and device for locating an emitting source of which measurements of emission propagation at locations different from that of the emitting source can be obtained from those locations, lacking space perception, using a sensor mobile along a self-generated path. After having obtained an emission propagation measurement from the mobile sensor at the mobile sensor location, a free energy variation for moving the sensor from its current location to each of plural possible next locations of the mobile sensor is computed, the free energy being computed as a function of a standardized projected probability field of the location of the diffusing source based on previous emission propagation measurements obtained along the self-generated path. A minimum free energy variation value amongst the computed free energy variations is determined and the location associated with the determined minimum free energy variation is identified as being the next location of the sensor.

FIELD OF THE INVENTION

The present invention relates generally to methods and systems forlocating emitting sources and more specifically to a method, a system,and a computer program for locating an emitting source of whichmeasurements of emission propagation at locations different from that ofthe emitting source can be obtained from those locations, lacking spaceperception.

BACKGROUND OF THE INVENTION

While insects are able to navigate in turbulent streams to find theirmates or food from sparse pheromone or odor detection, even in case ofpoor space perception for some of them, machines face difficulties inhandling such a problem despite the needs e.g., locating drugs, chemicalleaks, explosives, and mines. Semiconductor gas-sensors are able todetect the presence or absence of specific odorous substances and todetermine their concentration. However, locating the sources has to takeinto account the environment and particularly the air or liquid flow inwhich the odorous substance diffuses in a chaotic way.

Most of the robots equipped with an odor sensor use the localconcentration gradient of an odorous substance to determine locally thedirection of its source, referred to as chemotactic search strategy.However, chemotactic search strategies based on local concentrationgradients require concentration to be sufficiently high so that itsaverage difference measured at two nearby locations is larger thantypical fluctuations. The signal-to-noise ratio depends of course on theaveraging time and might be improved by waiting. However, averageconcentration may be decaying rapidly e.g., exponentially, with thedistance away from the source and in this weak signal-to-noise (dilute)case waiting becomes worse than exploratory motion. As an illustration,FIG. 1 depicts an example of an environment where an odorous substanceis diffused within the atmosphere and where odorous substanceconcentration cannot be used locally to determine the odorous substancesource.

Chemotaxis requires a reliable measurement of local gradients. This isnot feasible for robots located far away from the source and severelylimits the range of application of automated source localization byrobots. Existing chemotaxic robots might take several minutes to locatea source a few meters away.

Therefore, there is a need to provide a method and systems for solvingthe challenge of locating an emitting source, e.g. locating particle ormolecule sources or heat sources in a dilute environment, in particularfor the design of sniffers or robots that track chemicals emitted bydrugs, chemical leaks, explosives and mines.

To solve these issues, there is provided an efficient method, device,and computer program to search for an emitting source in a turbulentflow, lacking space perception and cognitive and/or probabilistic maps.

SUMMARY OF THE INVENTION

Faced with these constraints, the inventors provide a method, a device,and a computer program for locating an emitting source of whichmeasurements of emission propagation at locations different from that ofthe emitting source can be obtained from those locations, in a highlyvariable environment, lacking space perception.

It is a broad object of the invention to remedy the shortcomings of theprior art as described above.

According to a first aspect of the invention there is provided a methodfor locating an emitting source of which a measurement of emissionpropagation at a different location from that of the emitting source canbe obtained from that different location, in a searching space, lackingspace perception, using a mobile sensor configured to obtain emissionpropagation measurements, the sensor being mobile along a self-generatedpath, the method comprising the steps of,

-   -   obtaining an emission propagation measurement from the mobile        sensor at the mobile sensor location;    -   for a plurality of possible next locations of the mobile sensor,        computing a free energy variation for moving the sensor from its        current location to each of the plurality of possible next        locations, the free energy being computed as a function of a        standardized projected probability field of the location of the        diffusing source based on previous emission propagation        measurements obtained along the self-generated path;    -   determining a minimum free energy variation value amongst the        computed free energy variations; and    -   identifying the location associated with the determined minimum        free energy variation as being the next location of the mobile        sensor.

Accordingly, the invention allows an emitting source such as a particleor molecule source or a heat source to be located in a diluteenvironment by a system having limited computing resources and lackingspace perception and cognitive and/or probabilistic maps.

In an embodiment, the free energy variation is computed as a function ofa first element that favors exploration of the searching space and as afunction of a second element that favors exploitation of previousemission propagation measurements obtained along the self-generatedpath, the first and second elements being weighted in relation to eachother.

In a particular embodiment, the method further comprises a step ofdetermining whether the current sensor location or the next location ofthe mobile sensor corresponds to the location of the emitting source.

The steps of obtaining an emission propagation measurement, computing afree energy variation, determining a minimum free energy variationvalue, identifying the next location of the mobile sensor, anddetermining whether the current sensor location or the next location ofthe mobile sensor corresponds to the location of the emitting source areadvantageously repeated until the location of the emitting source islocated.

In a particular embodiment, the method further comprises a step ofcomparing the determined minimum free energy variation value with apredetermined threshold, the next location of the mobile sensor beingidentified as the location associated with the determined minimum freeenergy variation or as the current sensor location as a function of theresult of the comparison.

In a particular embodiment, the method further comprises a step ofstoring the obtained emission propagation measurement in memory alongwith the current mobile sensor location.

In a particular embodiment, the method further comprises a step ofmoving the mobile sensor to the identified next location of the mobilesensor.

In a particular embodiment, a measurement of emission propagationcharacterizes detecting or not detecting emission of the emittingsource.

According to a particular embodiment, the standardized projectedprobability field of the location of the diffusing source, representingthe probability P_(t) ^(M)({right arrow over (r)}₀|Θ_(t)) that theemitting source is located at location {right arrow over (r)}₀, knowingpath Θ_(t), can be expressed with the following formula:

${P_{t}^{M}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)} = \frac{e^{\frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{G}}}^{2}}{\lambda_{G}^{2}}}\left( {1 - {\frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}\; e^{\frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{j}}}^{2}}{\lambda_{u}^{2}}}}}} \right)}{Z_{t}}$where

{right arrow over (r)}_(G) represents a damped mass center of previouslocations where emission of the emitting source has been detected andλ_(G) represents a scale associated with a Gaussian term approximatingterms associated with detection of emission of the emitting source;

{right arrow over (r)}_(j) represents the locations of the searchingsystem sensor where emission of the emitting source has not beendetected;

N_(M) is the number of locations where measurements have been obtained;

λ_(u) represents a scale of a Gaussian term approximating termsassociated with absence of detection of emission of the emitting source;and

Z_(t) is a normalization constant.

In an embodiment, the step of obtaining an emission propagationmeasurement from the mobile sensor at the mobile sensor location furthercomprises the step of obtaining at least an emission propagationmeasurement from at least a second mobile sensor, different from themobile sensor, at the location of the at least a second mobile sensor,the at least a second mobile sensor being configured to obtain emissionpropagation measurements, the at least a second mobile sensor beingmobile along at least a second self-generated path, the free energybeing computed as a function of the standardized projected probabilityfield of the location of the diffusing source based on previous emissionpropagation measurements obtained along the self-generated path and onat least a previous emission propagation measurement obtained along theat least a second self-generated path.

In an embodiment, the standardized projected probability field of thelocation of the diffusing source, representing the probability P_(t)^(M)({right arrow over (r)}₀|Θ_(t)) that the emitting source is locatedat location {right arrow over (r)}₀, knowing path Θ_(t), can beexpressed with the following formula:

${P_{t}^{M}\left( {\overset{\rightarrow}{r_{0}}❘\Theta_{t}} \right)} = \frac{e^{- \frac{{{\overset{\rightarrow}{r_{0}} - \overset{\rightarrow}{r_{G}^{N_{S}}}}}^{2}}{\lambda_{G}^{2}}}\left( {{- \frac{1}{N_{M}}}{\sum\limits_{k = 1}^{N_{S}}\;{\sum\limits_{j = {N_{t} - {\lbrack\frac{({N_{M} + 1})}{N_{S}}\rbrack}}}^{N_{S}}\; e^{- \frac{{{\overset{\rightarrow}{r_{0}} - \overset{\rightarrow}{r_{j}^{k}}}}^{2}}{\lambda_{u}^{2}}}}}} \right)}{Z_{t}}$where

{right arrow over (r_(G) ^(N) ^(s) )} represents a damped mass center ofprevious locations where emission of the emitting source has beendetected by the mobile sensor and the at least a second mobile sensorand λ_(G) represents a scale associated with a Gaussian termapproximating terms associated with detection of emission of theemitting source;

{right arrow over (r_(j) ^(k))} represents the locations of a searchingsystem sensor having index k where emission of the emitting source hasnot been detected;

N_(M) is the number of locations where measurements have been obtained;

λ_(u) represents a scale of a Gaussian term approximating termsassociated with absence of detection of emission of the emitting source;and

Z_(t) is a normalization constant.

The emitting source may be a source of particles, molecules, orfragments of molecules, a heat source, or a source of data (the sensorbeing responsive to data patterns).

A second aspect of the invention provides a computer-readable storagemedium storing instructions of a computer program for implementing themethod in accordance with any embodiment of the first aspect of theinvention.

A third aspect of the invention provides an apparatus or a set ofapparatus comprising means adapted for carrying out each step of themethod in accordance with any embodiment of the first aspect of theinvention.

In an embodiment, the apparatus or the set of apparatus furthercomprises first means embedding said mobile sensor, said first meansbeing mobile in said search space, and second means for computing thefree energy variation, determining a minimum free energy variationvalue, and identifying the next location of the mobile sensor, the firstand second means comprising means for exchanging data with each other.

Since the present invention can be implemented in software, the presentinvention can be embodied as computer readable code for provision to aprogrammable apparatus on any suitable carrier medium. A tangiblecarrier medium may comprise a storage medium such as a floppy disk, aCD-ROM, a hard disk drive, a magnetic tape device or a solid statememory device and the like. A transient carrier medium may include asignal such as an electrical signal, an electronic signal, an opticalsignal, an acoustic signal, a magnetic signal or an electromagneticsignal, e.g. a microwave or RF signal.

BRIEF DESCRIPTION OF THE DRAWINGS

Further advantages of the present invention will become apparent tothose skilled in the art upon examination of the drawings and detaileddescription. It is intended that any additional advantages beincorporated herein.

Embodiments of the invention will now be described, by way of exampleonly, and with reference to the following drawings in which:

FIG. 1 illustrates an example of an environment where an odoroussubstance is diffused within the atmosphere and where odorous substanceconcentration cannot be used locally to determine the odorous substancesource;

FIG. 2 illustrates an example of the architecture of a searching systemaccording to a particular embodiment;

FIG. 3 illustrates steps of an algorithm to search for an emittingsource of which measurements of emission propagation at locationsdifferent from that of the emitting source can be obtained from thoselocations, in a highly variable environment, lacking space perception,according to an embodiment; and

FIG. 4, comprising FIGS. 4a and 4b , illustrates an advantage of usingseveral robots to locate one or several emitting sources.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

According to a particular embodiment, an efficient method, device, andcomputer program is provided to search for an emitting source in aturbulent flow, lacking space perception and cognitive and/or orprobabilistic maps. The method, device, and computer program alloweffective searches, in complex varying environments, with limited accessto cues, by using a searching system having limited space processingcapacities. The emitting source can be, for example, a source ofparticles, molecules, or fragments of molecules, a heat source, or anyother source of which measurements of emission propagation at locationsdifferent from that of the emitting source can be obtained from thoselocations. The emitting source can also be a source of data or a sourceof errors in a communication network.

According to an embodiment, the searching system comprises a processingdevice, a mobile sensor, and moving means such as motorized wheels. Allthe elements of the searching system can be integrated in one movingapparatus, like an active mobile robot (that is to say an “intelligent”mobile robot), or can be integrated as two or more apparatuscommunicatively coupled. For the sake of illustration, the searchingsystem may comprise a static computer, for example of the PC (PersonalComputer) type, provided with a communication interface, and a passivemobile robot (that is to say a remotely controlled mobile robot)comprising a sensor, a communication interface that is compatible withthat of the computer, and motorized wheels controlled by the computer.The communication interfaces may comply with any communication standard,preferably a wireless communication standard such as the one known asISO/CEI 8802-11. The active or passive mobile robot can move along aself-generated path, with or without user-defined constraints, to locatethe diffusing source.

FIG. 2 illustrates an example of the architecture of a searching system200. As shown, the searching system 200 comprises a central processingunit (CPU) 205, a Read-Only Memory (ROM) 210, a Random Access Memory(RAM) 215, and an input/output (I/O) subsystem 220, all of them beingconnected to a system bus 225. The I/O subsystem 220 may include one ormore controllers for input/output devices such as user interface 230(which may comprise a keyboard, a cursor control device, and a displaydevice), communication interface 235 (which can be of wire or wirelesstype), mass storage device 240, at least one sensor 245 for remotelyobtaining emission propagation measurements of the emitting sourcessearched for, and actuator controller 250 for controlling displacementsof the searching system (or at least of the sensor 245).

Depending upon the application of the searching system 200, one or morefurther I/O devices may be connected to the I/O subsystem 220.Typically, the hardware system 200 is controlled by an operating systemthat can be stored in ROM 210 or in mass storage device 245, which inturn controls various tools and applications that are generally loadedin RAM 215.

The searching system may include several types of sensors to localizeseveral kinds of emitting sources and/or to analyze particular featuresof the environment e.g., detecting obstacles.

Some of the elements of the searching system such as CPU 205, ROM 210,RAM 215, I/O subsystem 220, all of them being connected to a system bus225, user interface 230, communication interface 235, and/or massstorage device 240 may be part of a generic computer device, handhelddevice, or any kind of computer device.

It should be noted that according to particular applications, the activeor passive mobile robot can be a virtual mobile robot, for example avirtual robot allowing a data source or an error source in acommunication network to be searched for. In such a case, the sensors tobe used are preferably sensors that are responsive to data patterns.

According to a particular embodiment, the searching system is based onthe use of the projection of a probability of the emitting sourceposition into a standardized form, to avoid the need of advanced spaceperception, and on the evaluation by the searching system (of which themobile sensor position is known from uncorrected noisy cueless pathintegration) of a free energy that is minimized in order to find theright path to the source. A parameter referred to as the “internaltemperature” allows an efficient balance between exploration of thesearch space and exploitation of information already obtained.

FIG. 3 illustrates steps of an algorithm to search for an emittingsource of which measurements of emission propagation at locationsdifferent from that of the emitting source can be obtained from thoselocations, in a highly variable environment, lacking space perception,according to an embodiment.

A first step (step 300) aims at determining a mapless search functionF_(t)(Θ_(t)) representing a free energy that is to be minimized in orderto find the right path to the source. An example of such a maplesssearch function is described herein below.

In a next step, a measurement of emission propagation related to theemitting source, at sensor location, is obtained. Such a measurement istypically a value, preferably a normalized value, for example a valuevarying from zero to one hundred, or a binary value, for example abinary value of which the value zero indicates that no emissionpropagation characteristic of the emitting source has been detected andthe value one indicates that an emission propagation characteristic ofthe emitting source has been detected. It is to be noted that a binaryvalue may be obtained from another type of value by using appropriatethresholds.

Next, according to a particular embodiment, a test is performed todetermine whether or not the emitting source is located at sensorlocation {right arrow over (r)}_(t) (step 310). Such a test can beperformed, for example, by comparing the measured value with a thresholdor by comparing a detection rate with a threshold.

Once a measurement of emission propagation related to the emittingsource, at sensor location {right arrow over (r)}_(t), has beenobtained, a set of next possible sensor locations {right arrow over(r)}_(t+dt) ^(i) is determined (step 315). According to a particularembodiment, a set of next possible sensor relative locations arepredetermined. For the sake of illustration, such a set of next possiblesensor relative locations can be set as follows:{right arrow over (r)}_(t+dt) ^(i)(x_({right arrow over (r)}) _(t)+1,y_({right arrow over (r)}) _(t) ){right arrow over (r)}_(t+dt) ²(x_({right arrow over (r)}) _(t)−1,y_({right arrow over (r)}) _(t) ){right arrow over (r)}_(t+dt) ³(x_({right arrow over (r)}) _(t),y_({right arrow over (r)}) _(t) +1){right arrow over (r)}_(t+dt) ⁴(x_({right arrow over (r)}) _(t),y_({right arrow over (r)}) _(t) −1){right arrow over (r)}_(t+dt) ⁵(x_({right arrow over (r)}) _(t)+1,y_({right arrow over (r)}) _(t) +1){right arrow over (r)}_(t+dt) ⁶(x_({right arrow over (r)}) _(t)−1,y_({right arrow over (r)}) _(t) +1){right arrow over (r)}_(t+dt) ⁷(x_({right arrow over (r)}) _(t)+1,y_({right arrow over (r)}) _(t) +1){right arrow over (r)}_(t+dt) ⁸(x_({right arrow over (r)}) _(t)−1,y_({right arrow over (r)}) _(t) −1)

Next, a variation ΔF_(t) of the free energy as defined by thepredetermined mapless search function F_(t)(Θ_(t)) between sensorlocation {right arrow over (r)}_(t) and each of the next possible sensorlocations {right arrow over (r)}_(t+dt) ^(i) is computed. The resulthaving the minimum value is selected to determine the next sensorlocation {right arrow over (r)}_(t+dt) (step 320).

According to a particular embodiment, the sensor may stay at itsprevious location {right arrow over (r)}_(t) if, for example, theminimum variation ΔF_(t) of the free energy exceeds a predeterminedthreshold.

After having being determined, the sensor is moved to the next sensorlocation {right arrow over (r)}_(t+dt) (step 325) and the algorithm isbranched to step 305 to obtain a new measurement of emission propagationrelated to the emitting source, at a sensor location, and to repeat theprevious described steps until the emitting source is reached. Theobtained measurement of emission propagation is advantageously memorizedalong with current sensor location {right arrow over (r)}_(t) to belater used for determining a direction to follow.

According to a particular embodiment, test 310 performed to determinewhether or not the emitting source is located at sensor location {rightarrow over (r)}_(t) may be replaced by test 310′ that aims atdetermining whether or not the emitting source is located at location{right arrow over (r)}_(t+dt). Such a test may consist in comparing thevalue of F_(t+dt)(Θ_(t)) with a predetermined threshold. As describedherein above, function F_(t)(Θ_(t)) can be chosen so thatF_(t+dt)(Θ_(t))=1 if the source is at location {right arrow over(r)}_(t+dt). Such a test may also be based on another type ofmeasurements.

As described above, the searching system moves along a self-generatedpath denoted Θ_(t) (with or without user-defined constraints) and, ateach iterating step, determines a direction to follow based on theevents experienced at previous location of path Θ_(t). The searchingsystem accesses its position from uncorrected path integration and isconsidered to evolve in a cue-limited environment preventing an accuratecorrection of its position. According to this particular embodiment,information bearing events are based on measurements of emissionpropagation of the emitting source at the sensor location. For the sakeof example, it can be based on either detections of particles (rarelocal events) or the absence of detection of particles (frequent lesslocalized events). Measurements of emission propagation can be comparedto one or more thresholds, for example if the measurements representparticle density, to result in detections or absences of detection.

It is to be noted that the environment may be characterized by afunction denoted R({right arrow over (r)}|{right arrow over (r)}₀)representing the rate of detection, which gives a rate of detection atlocation {right arrow over (r)} when the source is located at location{right arrow over (r)}₀. A simplified model of such a function R({rightarrow over (r)}|{right arrow over (r)}₀) can be, for example, a simpleGaussian function. In the case where the diffusing source is a particlediffusing source in the air, the exact expression reads as follows:

${R\left( {\overset{\rightarrow}{r}❘{\overset{\rightarrow}{r}}_{0}} \right)} = {\frac{R}{\ln\left( \frac{\lambda}{a} \right)}e^{- {(\frac{{({y - y_{0}})}V}{2\; D})}}{K_{0}\left( \frac{{\overset{\rightarrow}{r} - {\overset{\rightarrow}{r}}_{0}}}{\lambda} \right)}}$where

-   -   τ and R represent the lifetime of emitted particles and their        emission rate, respectively;    -   {right arrow over (V)} represents a mean wind in the        environment;    -   a is the characteristic length of the particle detector;    -   K₀ is the modified Bessel function of order zero;    -   (y and y₀) represents where the wind was taken to blow in the y        direction and y₀ is the y-coordinate of the diffusing source;        and

λ is defined as follows:

$\lambda = \sqrt{\frac{D\;\tau}{1 + \frac{V^{2}\tau}{4\; D}}}$where D represents the coefficient of diffusion of the particles.

The absence of complex space perception prevents the direct decoding byinference of the information stored in the path Θ_(t) using Bayesianinference. Hence, all terms involved in the path choosing mechanism haveto be expressed in a standardized form to be computed without thecomplete expression of the time varying probability of finding thesource at a particular location. The following standardized projectedprobability can be used,

${P_{t}^{M}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)} = \frac{e^{\frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{G}}}^{2}}{\lambda_{G}^{2}}}\left( {1 - {\frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}\; e^{\frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{j}}}^{2}}{\lambda_{u}^{2}}}}}} \right)}{Z_{t}}$where

-   -   P_(t) ^(M)({right arrow over (r)}₀|Θ_(t)) is the probability        that the emitting source is located in {right arrow over (r)}₀        knowing path    -   Θ_(t),    -   {right arrow over (r)}_(G) represents the damped mass center of        detections and λ_(G) represents the scale associated with the        Gaussian term approximating the detection terms;    -   {right arrow over (r)}_(j) represents the locations of the        searching system sensor where no detection has been made;    -   N_(M) is the number of measurement locations;    -   λ_(u) represents the scale of the Gaussian terms approximating        the non-detection terms; and    -   Z_(t) is a normalization constant.

The lack of space perception coupled with the accumulation ofpositioning errors lead, preferably, to precautions being taken inrelation to the amount of “memory” that the searching system should keepfrom its past. Hence, the damped center of mass can be expressed asfollows:

${\overset{\rightarrow}{r}}_{G} = \frac{\sum\limits_{i = 1}^{G}\;{\gamma^{G - i}{\overset{\rightarrow}{r}\left( t_{i} \right)}}}{\sum\limits_{i = 1}^{G}\;\gamma^{G - i}}$where γ<1 and {t_(i),i=1 . . . G} represents the times of themeasurements.

As described above, the searching system is based on the use of theprojection of a probability of the diffusion source position into astandardized form and on the evaluation of a free energy. Accordingly,to decide the direction to be taken after experiencing theself-generated path Θ_(t) (in view of user-defined constraints if any),a function of P_(t) ^(M)({right arrow over (r)}₀|Θ_(t)) is to bedetermined. The minimum value obtained from such a function of P_(t)^(M)({right arrow over (r)}₀|Θ_(t)) according to one of a plurality ofpossible directions, is used to choose the direction to be taken.

Optimization based on entropy computation is known to be efficient (inparticular, from EP 1,881,389). However, since it has been observed thatentropy tends to shift the exploration versus exploitation balancetowards exploration, another term has been added to reinforce maximumlikelihood behavior and so shift the balance towards exploitation.Accordingly, a free energy can be expressed according to the followingformula:F _(t)(Θ_(t))=W _(t)(Θ_(t))+TS(Θ_(t))where

-   -   W_(t)(Θ_(t)) represents a working energy;    -   S(Θ_(t)) represents the Shannon entropy; and    -   T a temperature parameter that controls the relative value        between these two elements.

Therefore, the free energy can be expressed as follows:

${F_{t}\left( \Theta_{t} \right)} = {{\underset{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{G}}} \leq \frac{\lambda_{G}}{2}}{\int\int}d{\overset{\rightarrow}{r}}_{0}{P_{t}^{M}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)}} + {T\left\lbrack {- {\int{\int{d{\overset{\rightarrow}{r}}_{0}{P_{t}^{M}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)}\log\;{P_{t}^{M}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)}}}}} \right\rbrack}}$

When moving from the sensor position at time t, denoted {right arrowover (r)}_(t), to a neighboring position corresponding to time t+dt,denoted {right arrow over (r)}_(t+dt), the expected variation of thefree energy is the sum of two terms, one accounting for the discovery ofthe source and the other one for its absence. This variation reads asfollows:

$\begin{matrix}{{\Delta\;{F_{t}\left( {\left. {\overset{\_}{r}}_{t}\rightarrow{\overset{\_}{r}}_{t + {dt}} \right.❘\Theta_{t}} \right)}} = {{{P_{t}^{M}\left( {\overset{\_}{r}}_{t + {dt}} \right)}\Delta\; F_{t}^{discovery}} + {\left( {1 - {P_{t}^{M}\left( {\overset{\_}{r}}_{t + {dt}} \right)}} \right)\Delta\; F_{t}^{nothing}}}} \\{= {{{P_{t}^{M}\left( {\overset{\_}{r}}_{t + {dt}} \right)}\left\lbrack {1 - {F_{t}\left( \Theta_{t} \right)}} \right\rbrack} + \left( {1 - {P_{t}^{M}\left( {\overset{\_}{r}}_{t + {dt}} \right)}} \right)}} \\{\left\lbrack {{{\rho_{0}\left( {\overset{\_}{r}}_{t + {dt}} \right)}\Delta\;{F_{t}^{0}\left( \Theta_{t} \right)}} + {{\rho_{1}\left( {\overset{\_}{r}}_{t + {dt}} \right)}\Delta\;{F_{t}^{1}\left( \Theta_{t} \right)}}} \right\rbrack}\end{matrix}$where

-   -   P_(t) ^(M)({right arrow over (r)}_(t+dt)) represents the        probability that the source is at position {right arrow over        (r)}_(t+dt);    -   F_(t) (Θ_(t)) represents the free energy of the searcher at time        t;    -   F_(t+dt)(Θ_(t))=1 is the free energy if the source is found;    -   [1−P_(t) ^(M)(Θ_(t))] is the probability that the source is not        at position {right arrow over (r)}_(t+dt);    -   ρ_(i)({right arrow over (r)}_(t+dt)) represents the expected        probability of having i detections; and    -   ΔF_(t) ^(i)(Θ_(t)) is the expected free energy variation if        there were i detections.

According to the Poisson detection function, the expected probability ofhaving i detections can be expressed as follows:

${\rho_{i}\left( \overset{\rightarrow}{r} \right)} = \frac{\left\lbrack \left( \overset{\rightarrow}{r} \right) \right\rbrack^{i}e^{- {h{(\overset{\rightarrow}{r})}}}}{i!}$where h({right arrow over (r)}) represents the expected average hit ratethat can be expressed as follows:h({right arrow over (r)})=∫d{right arrow over (r)} ₀ R({right arrow over(r)}|{right arrow over (r)} ₀)P _(t) ^(M)({right arrow over (r)} ₀)where R({right arrow over (r)}|{right arrow over (r)}₀) is the rate ofdetection of the searching system if it is at the position and thesource is at {right arrow over (r)}₀. This can be approximated using aGaussian function.

The updating rule of {right arrow over (r)}_(G), to avoid storing thehit positions, can be expressed as follows:

${\overset{\rightarrow}{r}}_{G + 1} = {{\gamma\frac{1 - \gamma^{G}}{1 - \gamma^{G + 1}}{\overset{\rightarrow}{r}}_{G}} + {\frac{1 - \gamma}{1 - \gamma^{G + 1}}{\overset{\rightarrow}{r}\left( t_{G + 1} \right)}}}$

In case of turbulent transport (e.g. wind), the updating procedure of{right arrow over (r)}_(G) can be expressed as follows:

$x_{G} = \frac{\sum\limits_{i = 1}^{G}{{\gamma(V)}^{G - i}{x\left( t_{i} \right)}}}{\sum\limits_{i = 1}^{G}{\gamma(V)}^{G - i}}$andy _(G) =y(t _(G))with the turbulent transport being oriented in the −y (minus y)direction (at time t), e.g. wind blowing in the −y direction, and γ(V)is the damping factor decreasing with rising velocity.

Depending on the characteristics of the environment where the emittingsource is to be found, the shape of γ(V) may be chosen to better fit thecharacteristics of the detection. Linear dependency with a maximumvelocity is found to be efficient. Obviously, if the turbulent transportdirection varies with a similar time scale as the sensor is moved, thesearch becomes extremely difficult, and the updating procedure of {rightarrow over (r)}_(G) is preferably the one used in the turbulentless(e.g. windless) scheme. It is to be noted that in order to ensure thatan emitting source will be found, the drift velocity information isunderexploited. The symmetry of the distribution ensures that thesearcher can go in the opposite direction to the turbulent transport(e.g. wind) if it has moved beyond the source position.

Drift velocity information processing can be improved as follows: aftereach detection, a random distance L is randomly generated from anexponential distribution of correlation length L_(T)=χλ_(e)(χ beinguser-defined) and the sensor is moved according to distance L in theopposite direction to the turbulent transport such as wind (at thetime). Conversely, to reinforce stability of the search for the rare(<1/500) searches where a sequence of detection with low probabilitieshappened (far from the source), the sensor is moved back to the lastpoint (with the accumulated odometry error) where a detection has beenmade after a user-defined time without any detection. Finally, initialspiraling is made on an asymmetrical spiral which is more extended whengoing in the opposite direction to the turbulent transport (e.g. wind)than when going in the direction of the turbulent transport.

If the turbulent transport is expected not to change direction (or tochange direction only slightly) during the search, asymmetry betweenturbulent transport direction and perpendicular direction can be addedto P_(t) ^(M)({right arrow over (r)}₀|Θ_(t)) to reinforce turbulenttransport information use.

If the constraint of always finding the source is removed, theefficiency of the search for the source can be improved. For example,reducing the choice of possible moving points to the ones in theopposite direction to the turbulent transport (e.g. wind) anddiminishing the internal temperature T to reinforce informationexploitation leads to fast searches when the searcher finds the source.

It is to be noted that all the terms involved in determining thedirection to be taken after experiencing the self-generated path Θ_(t),i.e. F_(t)(Θ_(t)), W_(t)(Θ_(t)), S(Θ_(t)), h_(t)({right arrow over(r)}_(t)), and ρ_(i)({right arrow over (r)}_(t)), can be approximated sothat they do not depend on P_(t) ^(M)({right arrow over (r)}₀|Θ_(t)),for example according to the following formulae:

${W\left( \Theta_{t} \right)} = {{\int{\int_{S}{d{\overset{\rightarrow}{r}}_{0}{P_{t}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)}}}} = {\frac{1}{Z_{t}}\left\lbrack {{\pi\left( {{{erf}\left( \frac{1}{2} \right)}\lambda_{G}} \right)}^{2} - {\frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M + 1}}}^{N_{t}}{\frac{{\pi\lambda}_{G}^{2}\lambda_{u}^{2}}{4\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)}e^{- \frac{{{\overset{\rightarrow}{r}}_{G} - r_{j}}}{({\lambda_{G}^{2} + \lambda_{u}^{2}})}}{\Phi\left( {x_{j},x_{G}} \right)}{\Phi\left( {y_{j},y_{G}} \right)}}}}} \right\rbrack}}$with S representing the integration domain that is defined here by

${{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{j}}} \leq \frac{\lambda}{2}$and where

${\Phi\left( {x_{j},x_{G}} \right)} = {{{erf}\left( \frac{{- \lambda_{d}^{2}} - {2x_{j}\lambda_{d}} + {2x_{G}\lambda_{d}} - \lambda_{u}^{2}}{2\lambda_{u}\sqrt{\lambda_{G}^{2} + \lambda_{u}^{2}}} \right)} - {{erf}\left( \frac{\lambda_{d}^{2} - {2x_{j}\lambda_{d}} + {2x_{G}\lambda_{d}} + \lambda_{u}^{2}}{2\lambda_{u}\sqrt{\lambda_{G}^{2} + \lambda_{u}^{2}}} \right)}}$${S\left( \Theta_{t} \right)} \approx {{\log\left( Z_{t} \right)} + {\frac{\pi}{2Z_{t}}\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)} + {\frac{\pi}{Z_{t}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}{e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - r_{j}}}^{2}}{({\lambda_{G}^{2} + \lambda_{u}^{2}})}}{\quad\left\lbrack {\frac{\lambda_{G}^{2} + \lambda_{u}^{2}}{\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)} - \frac{\lambda_{G}{\lambda_{u}\left( {{\lambda_{G}^{3}{\lambda_{u}\left( {x_{G} - x_{i}} \right)}^{2}} + {\lambda_{G}{\lambda_{u}^{3}/2}}} \right)}}{\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)} - x}\rightarrow y \right\rbrack}}}}}$with the approximation

${\log\left( {1 - {\frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}e^{- \frac{{{{\overset{\rightarrow}{r}}_{0} - r_{j}}}^{2}}{\lambda_{u}^{2}}}}}} \right)} \approx {{- \frac{1}{N_{M}}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}e^{{- \frac{{{{\overset{\rightarrow}{r}}_{0} - r_{j}}}^{2}}{\lambda_{u}^{2}}}\;}}}$and where all the cross terms

$\frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}{e^{{- \frac{{{{\overset{\rightarrow}{r}}_{0} - r_{j}}}^{2}}{\lambda_{u}^{2}}}\;} \times \frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}e^{{- \frac{{{{\overset{\rightarrow}{r}}_{0} - r_{j}}}^{2}}{\lambda_{u}^{2}}}\;}}}}$have been neglected (mean field approximation); and

${h\left( \overset{\rightarrow}{r} \right)} = {\frac{\pi\kappa}{Z_{t}}\left( {{\frac{1}{2}\lambda_{G}^{2}e^{{- \frac{{{{\overset{\rightarrow}{r}}_{G} - r_{j}}}^{2}}{2\lambda_{u}^{2}}}\;}} - {\frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}{\frac{\lambda_{G}^{2} + \lambda_{u}^{2}}{\left( {\lambda_{G}^{2} + {2\lambda_{u}^{2}}} \right)}{\Psi\left( {x_{G},x_{j},x} \right)}{\Psi\left( {y_{G},y_{j},y} \right)}}}}} \right)}$with the user defined factor κ and

${\Psi\left( {x_{G},x_{j},x} \right)} = {e^{\frac{({{{- {({x_{G}^{2} + x^{2}})}}{({\lambda_{G}^{2} + \lambda_{u}^{2}})}} + {\lambda_{G}^{2}{({{{- 2}x_{j}^{2}} + {2x_{G}x_{j}} + {2{xx}_{j}}})}} + {2x_{G}x\;\lambda_{u}^{2}}})}{{({\lambda_{G}^{2} + {2\lambda_{u}^{2}}})}\lambda_{G}^{2}}}.}$

It is also to be noted that such formulae can be expressed differentlyto optimize updates and to reduce required memory accesses. To that end,the standardized form of the probability field can be expressed asfollows:

${P_{t}^{M}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)} = {{\frac{e^{- \frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{G}}}^{2}}{\lambda_{G}^{2}}}\left( {1 - {\sum\limits_{j = 1}^{N_{t}}{\rho^{N_{t} - j}e^{- \frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{j}}}^{2}}{\lambda_{G}^{2}}}}}} \right)}{Z_{t}}\mspace{14mu}{with}\mspace{14mu}\rho} < 1}$

By defining Ξ_(t)({right arrow over (r)}₀|Θ_(t)) as follows:

${\Xi_{t}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)} = {\sum\limits_{j = 1}^{N_{t}}{\rho^{N_{t} - j}e^{- \frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{j}}}^{2}}{\lambda_{u}^{2}}}}}$an updating rule may be expressed as follows:

${\Xi_{t + {dt}}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t + {dt}}} \right)} = {{{\rho\Xi}_{t}\left( {{\overset{\rightarrow}{r}}_{0}❘\Theta_{t}} \right)} + e^{- \frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{N + 1}}}^{2}}{\lambda_{u}^{2}}}}$Similarly by defining Z_(t) as follows:Z _(t)=πλ_(G) ²−Γ_(t)with

$\Gamma_{t} = {\pi\frac{\lambda_{G}^{2}\lambda_{u}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}{\sum\limits_{j = 1}^{N_{t}}{\rho^{N_{t} - j}e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - {\overset{\rightarrow}{r}}_{j}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}}}}$and the following updating rules

$\Gamma_{t + {d\; t}} = {{\rho\Gamma}_{t + {d\; t}} + {\pi\frac{\lambda_{G}^{2}\lambda_{u}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - {\overset{\rightarrow}{r}}_{N + 1}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}}}$henceZ _(t+dt)=πλ_(G) ²−Γ_(t+dt).Furthermore, the work energy becomes

${W\left( \Theta_{t} \right)} = {\frac{1}{Z_{t}}\left\lbrack {{\pi\left( {{{erf}\left( \frac{1}{2} \right)}\lambda_{G}} \right)}^{2} - {\sum\limits_{j = 1}^{N_{t}}{\rho^{N_{t} - j}\frac{{\pi\lambda}_{G}^{2}\lambda_{u}^{2}}{4\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)}e^{- \frac{{{{\overset{\rightarrow}{r}}_{0} - {\overset{\rightarrow}{r}}_{G}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}{\Phi\left( {x_{j},x_{G}} \right)}{\Phi\left( {y_{j},y_{G}} \right)}}}} \right\rbrack}$By defining

$\Omega_{t} = {\sum\limits_{j = 1}^{N_{t}}{\rho^{N_{t} - j}\frac{{\pi\lambda}_{G}^{2}\lambda_{u}^{2}}{4\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)}e^{- \frac{{{{\overset{\rightarrow}{r}}_{j} - {\overset{\rightarrow}{r}}_{G}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}{\Phi\left( {x_{j},x_{G}} \right)}{\Phi\left( {y_{j},y_{G}} \right)}}}$and the updating rule

$\left. \Omega_{t + {dt}}\leftarrow{{\rho\Omega}_{t} + {\frac{{\pi\lambda}_{G}^{2}\lambda_{u}^{2}}{4\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)}e^{- \frac{{{{\overset{\rightarrow}{r}}_{N + 1} - {\overset{\rightarrow}{r}}_{G}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}{\Phi\left( {x_{N + 1},x_{G}} \right)}{\Phi\left( {y_{N + 1},y_{G}} \right)}}} \right.$hence

$\mspace{79mu}{{W\left( \Theta_{t + {dt}} \right)} = {\frac{1}{Z_{t + {dt}}}\left\lbrack {{\pi\left( {{{erf}\left( \frac{1}{2} \right)}\lambda_{G}} \right)}^{2} - \Omega_{t + {dt}}} \right\rbrack}}$and${S\left( \Theta_{t} \right)} \approx {{\log\left( Z_{t} \right)} + {\frac{\pi}{2\; Z_{t}}\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)} + {\frac{\pi}{Z_{t}}{\sum\limits_{i = 1}^{N_{t}}{\rho^{N_{t} - i}{e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - {\overset{\rightarrow}{r}}_{i}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}\left\lbrack {\frac{\lambda_{G}^{2}\lambda_{u}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}} - \frac{\lambda_{G}{\lambda_{u}\left( {{\lambda_{G}^{3}{\lambda_{u}\left( {x_{G} - x_{i}} \right)}^{2}} + {\lambda_{G}{\lambda_{u}^{3}/2}}} \right)}}{\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)^{2}} - \frac{\lambda_{G}{\lambda_{u}\left( {{\lambda_{G}^{3}{\lambda_{u}\left( {y_{G} - y_{i}} \right)}^{2}} + {\lambda_{G}{\lambda_{u}^{3}/2}}} \right)}}{\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)^{2}}} \right\rbrack}}}}}$with$\Psi_{t} = {\sum\limits_{i = 1}^{N_{t}}{\rho^{N_{t} - i}{e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - {\overset{\rightarrow}{r}}_{i}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}\left\lbrack {\frac{\lambda_{G}^{2}\lambda_{u}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}} - \frac{\lambda_{G}{\lambda_{u}\left( {{\lambda_{G}^{3}{\lambda_{u}\left( {x_{G} - x_{i}} \right)}^{2}} + {\lambda_{G}{\lambda_{u}^{3}/2}}} \right)}}{\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)^{2}} - \frac{\lambda_{G}{\lambda_{u}\left( {{\lambda_{G}^{3}{\lambda_{u}\left( {y_{G} - y_{i}} \right)}^{2}} + {\lambda_{G}{\lambda_{u}^{3}/2}}} \right)}}{\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)^{2}}} \right\rbrack}}}$the updating rule

$\Psi_{t + {dt}} = {{\rho\Psi}_{t + {dt}} + {e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - {\overset{\rightarrow}{r}}_{N + 1}}}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}}}\left\lbrack {\frac{\lambda_{G}^{2}\lambda_{u}^{2}}{\lambda_{G}^{2} + \lambda_{u}^{2}} - {\Theta\left( {x_{G},x_{N + 1}} \right)} - {\Theta\left( {y_{G},y_{N + 1}} \right)}} \right\rbrack}}$with$\mspace{20mu}{{\Theta\left( {x_{G},x_{N + 1}} \right)} = \frac{\lambda_{G}{\lambda_{u}\left( {{\lambda_{G}^{3}{\lambda_{u}\left( {x_{G} - x_{N + 1}} \right)}^{2}} + {\lambda_{G}{\lambda_{u}^{3}/2}}} \right)}}{\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)^{2}}}$leading to

${S\left( \Theta_{t + {dt}} \right)} \approx {{\log\left( Z_{t + {dt}} \right)} + {\frac{\pi}{2\; Z_{t + {dt}}}\left( {\lambda_{G}^{2} + \lambda_{u}^{2}} \right)} + {\frac{\pi}{Z_{t + {dt}}}{\Psi_{t + {dt}}.}}}$

It is to be noted that for both Ω_(t) and Ψ_(t) an approximation ispreferably made in the updating rule: the function Φ depends on {rightarrow over (r)}_(G) which varies in time but the updating rules assumethat {right arrow over (r)}_(G) does not vary for the terms earlier intime.

Finally,

$\mspace{20mu}{{h_{t}\left( \overset{\rightarrow}{r} \right)} = {\frac{\pi\kappa}{Z_{t}}\left( {{\frac{1}{2}\lambda_{G}^{2}e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - \overset{\rightarrow}{r}}}^{2}}{2\lambda_{G}^{2}}}} - A_{t}} \right)}}$with$\mspace{20mu}{A_{t} = {\sum\limits_{j = 1}^{N_{t}}{\rho^{N_{t} - j}\frac{\lambda_{G}^{2}\lambda_{u}^{2}}{{2\lambda_{u}^{2}} + \lambda_{G}^{2}}{\Omega\left( {x_{G},x,x_{N + 1}} \right)}{\Omega\left( {y_{G},y,y_{N + 1}} \right)}}}}$with${\Omega\left( {x_{G},x,x_{N + 1}} \right)} = e^{\frac{\lbrack{{- {x_{G}^{2}{({\lambda_{u}^{2} + \lambda_{G}^{2}})}}} - {x^{2}{({\lambda_{u}^{2} + \lambda_{G}^{2}})}} + {\lambda_{G}^{2}{({{{- 2}x_{N + 1}^{2}} + {2x_{G}x_{N + 1}} + {2{xx}_{N + 1}}})}} + {2x_{G}x\;\lambda_{u}^{2}}}\rbrack}{{({{2\lambda_{u}^{2}} + \lambda_{G}^{2}})} + \lambda_{G}^{2}}}$and the following updating rule

$A_{t + {dt}} = {{\rho\; A_{t}} + {\frac{\lambda_{G}^{2}\lambda_{u}^{2}}{{2\lambda_{u}^{2}} + \lambda_{G}^{2}}{\Omega\left( {x_{G},x,x_{N + 1}} \right)}{\Omega\left( {y_{G},y,y_{N + 1}} \right)}}}$leading to

$h_{t + {dt}} = {\frac{\pi\kappa}{Z_{t + {dt}}}\left( {{\frac{1}{2}\lambda_{G}^{2}e^{- \frac{{{{\overset{\rightarrow}{r}}_{G} - \overset{\rightarrow}{r}}}^{2}}{2\lambda_{G}^{2}}}} - A_{t + {dt}}} \right)}$

Accordingly, the direction to be taken can be computed easily withouthuge resources.

During the search, two processes ought to be balanced: exploration ofthe environment to get more information on the localization of theemitting source and exploitation of the already accumulated information.

In Infotaxis, as disclosed in particular in EP 1,881,389, the entropy isused to balance exploration and exploitation. Based on the reliableinference of the source position distribution P_(t)({right arrow over(r)}₀), the balance is efficiently managed with a clear advantagetowards exploration.

According to a particular embodiment, information accessible from P_(t)^(M)({right arrow over (r)}₀|Θ_(t)) is only partially reliable since thesearching system has no exact access to the sensor position and thedetection process may not correspond to the one used in the model used.The use of the working energy W_(t)(Θ_(t)) allows reinforcement of themaximum likelihood behavior and the temperature T allows the balance tobe shifted between exploration and exploitation. It is to be noted thatboth exploration and exploitation are present in the entropy S(Θ_(t))where mostly exploitation is present in the working energy W_(t)(Θ_(t)).This allows the temperature to actively control the balance.Furthermore, the working energy W_(t)(Θ_(t)) prevents the self-trappingbehavior observed during simulations performed for high temperatures(T>5), when a large number of detections has been made far from (rare)and close to the source.

It is to be noted that at short initial distances (between the sensorand the emitting source), the use of a low temperature value is moreefficient because the working energy W_(t)(Θ_(t)), which favors maximumlikelihood behavior, has more influence on the decision process than athigh temperature. As the initial distance rises, the optimum temperaturerises with a flattening of the mean search time-temperature curve.

According to a particular embodiment, the sensor is moved after aninitial detection. The initial movement is preferably defined as anArchimedean spiral. Such an Archimedean spiral movement is preferablyused until a second detection is made. In case of turbulent transport,the initial spiral is preferably asymmetrical, with the spiral beingwider in the opposite direction to the turbulent transport than in itsdirection.

Still according to a particular embodiment, a linear motion in a randomdirection may be chosen if the algorithm results in a decision of notmoving the sensor.

Still in a particular embodiment, the memory of the searching systemthat is used to store previous detections may be reset when numerousdetections have been made, after a user-defined time. Such an embodimentis particularly adapted to the case in which the sensor is initiallylocated far from the source and hence reaches the vicinity of the sourcewith numerous detections. Memory reset allows the balance ofexploration/exploitation to shift back to exploration.

According to a particular embodiment, several independent mobilesensors, for example several intelligent mobile robots, are used tolocate one or several emitting sources. The architecture and thefeatures of these mobile sensors may be similar to those describedabove.

If the mobile sensors are still unable to determine their positionaccurately in the searching space and cannot correct odometry error,they are provided with communication means so as to transmit and/orreceive data, in particular measurements of emission propagation and/orrelative position information. Such data can be transmitted and/orreceived on a regular basis whatever the relative position of robots isor on a regular basis only if the robots are close to each other.

For the sake of illustration, two robots can locate each other usingultrasound sensors, sonars, cameras, and the like. The detection can bemade by using local sensors or via a central computer able tocommunicate with the robots. Alternatively, all the robots are providedwith their mutual initial positions, these positions being thereforeknown at least approximately by each robot.

According to this embodiment, the mobile sensors (or robots) move alongself-generated paths Θ_(i) ^(t) i∈(1 . . . Ns), with N_(s) representingthe number of robots. The latter have to decide their future paths basedon information accumulated into paths Θ_(i) ^(t). More precisely, eachof the robots, moving in a medium where localization (SLAM, SimultaneousLocalization And Mapping) is either difficult or impossible, takesindividual decisions based on a global path Θ_(t) and estimates itsposition from uncorrected path integration. As a consequence, all robotsaccumulate odometry errors as they search for the emitting source(s) andhave no direct way to correct them.

The standardized projected probability that an emitting source islocated in {right arrow over (r)}₀ knowing path Θ_(t) can be defined asfollows:

${P_{t}^{M}\left( {\overset{\rightarrow}{r_{0}}❘\Theta_{t}} \right)} = \frac{e^{- \frac{{{\overset{\rightarrow}{r_{0}} - \overset{\rightarrow}{r_{G}^{N_{s}}}}}^{2}}{\lambda_{G}^{2}}}\left( {{- \frac{1}{N_{M}}}{\sum\limits_{k = 1}^{N_{s}}{\sum\limits_{j = {N_{t} - {\lbrack\frac{({N_{M} + 1})}{N_{s}}\rbrack}}}^{N_{s}}e^{- \frac{{{\overset{\rightarrow}{r_{0}} - \overset{\rightarrow}{r_{j}^{k}}}}^{2}}{\lambda_{u}^{2}}}}}} \right)}{Z_{t}}$where

-   -   {right arrow over (r_(G) ^(N) ^(S) )} represents the damped mass        center of detections made by all the robots and λ_(G) represents        the scale associated with the Gaussian term approximating the        detection terms;    -   {right arrow over (r_(j) ^(k))} represents the locations of the        robot having index k (i.e. the searching system sensor having        index k) where no detection has been made;    -   N_(M) is the number of measurement locations;    -   λ_(u) represents the scale of the Gaussian terms approximating        the non-detection terms; and    -   Z_(t) is a normalization constant.

Again, it is noted that the lack of space perception coupled with theaccumulation of positioning errors lead, preferably, to precautionsbeing taken in relation to the amount of “memory” that the searchingsystem should keep from its past.

Like the damped centre of mass expressed for the detections made by asingle robot, the damped mass center of detections made by all therobots can be expressed as follows:

$\overset{\rightarrow}{r_{G}^{N_{s}}} = \frac{\sum\limits_{i = 1}^{G}{\gamma^{G - i}\overset{\rightarrow}{r\left( t_{l} \right)}}}{\sum\limits_{i = 1}^{G}\gamma^{G - i}}$where γ<1, {right arrow over (r(t_(i)) )} is the position of one robotwhich made the detection at time t_(i), and {t_(i),i=1 . . . G}represents the times of the measurements ordered from the most recentdetections to the oldest detections.

In such a swarm searching process, robots share information regardingtheir trajectories. More precisely, the robots share informationregarding noisy trajectories that are determined without odometry errorcorrections (true positions may significantly differ from those that areshared).

If robots cannot share information on a regular basis (e.g. if robotsare far away from each other), the standardized projected probabilityP_(t) ^(M)({right arrow over (r₀)}|Θ_(t)) is updated as a function ofthe damped mass center of detections made by all the robots and oflocations of the robot having index k where no detection has been madeif such information is available (i.e. when robots can exchangeinformation) or as a function of the damped mass center of detectionsmade by the robot and of its locations where no detection has been made(i.e. when robots cannot exchange information).

To determine its next position, each robot minimizes the free energyF_(t)(Θ_(t)) as described above. As mentioned above, each robot takesindividual decisions to generate its own path and information is sharedthrough the part of the path that each robot shares.

FIG. 4, comprising FIGS. 4a and 4b , illustrates an advantage of usingseveral robots to locate one or several emitting sources.

FIG. 4a illustrates the variation of the average searching time as afunction of the average distance between the robots and the centers oftwo emitting sources (which are distant from each other by distance λ=30a.u.) and as a function of the number of robots used. Curve 600represents the variation of the average searching time when two robotsare used and curve 605 represents the variation of the average searchingtime when seven robots are used.

FIG. 4b illustrates the variation of the average searching time as afunction of the number of robots and as a function of the distancebetween the robots and the centers of two emitting sources (which aredistant from each other by distance λ=30 a.u.). Curve 610 represents thevariation of the average searching time when the distance between therobots and the centers of two emitting sources is equal to 75 a.u. andcurve 615 represents the variation of the average searching time whenthe distance between the robots and the centers of two emitting sourcesis equal to 45 a.u.

Embodiments of the invention may be used for robotic searches in randomenvironments, especially when localization is either unachievable or toocomplicated to be achieved continuously. This could also be named blindrobot searches. According to specific wording, embodiments allowsearches when SLAM is difficult to achieve.

Embodiments may also be used for searches in complex environments withsparse unreliable signals. For example searches for planes or boats inoceans by using detections of a remaining signal sporadically emittedand the local bathymetry.

Embodiments of the invention also find direct applications inreinforcement learning and machine learning. Examples of suchapplications concern search problems where a balance has to be foundbetween exploration and exploitation. Other applications are linked toquantitative economy that can be expressed, in particular, as treedecision problems where a balance between exploration and exploitationis to be achieved.

Depending on the applications, user-defined constraints can beassociated with the self-generated path. For the sake of illustration,if the system of the invention is used for locating a signal sourcealong wires, the self-generated path is limited to the wire network.

Naturally, in order to satisfy local and specific requirements, a personskilled in the art may apply to the solution described above manymodifications and alterations all of which, however, are included withinthe scope of protection of the invention as defined by the followingclaims.

The invention claimed is:
 1. A method for locating an emitting source ofwhich a measurement of emission propagation at a different location fromthat of the emitting source can be obtained from that differentlocation, in a searching space, lacking space perception, using a mobilesensor configured to obtain emission propagation measurements, thesensor being mobile along a self-generated path, the method comprising:obtaining an emission propagation measurement from the mobile sensor atthe mobile sensor location; for a plurality of possible next locationsof the mobile sensor, computing a free energy variation for moving thesensor from its current location to each of the plurality of possiblenext locations, the free energy being computed as a function of astandardized projected probability field of the location of the emittingsource based on previous emission propagation measurements obtainedalong the self-generated path without space perception; determining aminimum free energy variation value amongst the computed free energyvariations; identifying the location associated with the determinedminimum free energy variation as being the next location of the mobilesensor; moving the mobile sensor to the identified next location of themobile sensor, wherein a measurement of emission propagationcharacterizes detecting or not detecting emission of the emitting sourceand wherein the standardized projected probability field of the locationof the emitting source, representing the probability P_(t) ^(M)({rightarrow over (r)}₀|Θ_(t)) that the emitting source is located at location{right arrow over (r)}₀, knowing path Θ_(t), can be expressed with thefollowing formula:${P_{t}^{M}\left( {{\overset{->}{r}}_{o}❘\Theta_{t}} \right)} = \frac{e^{- \frac{{{{\overset{->}{r}}_{o} - {\overset{->}{r}}_{G}}}^{2}}{\lambda_{G}^{2}}}\left( {1 - {\frac{1}{N_{M}}{\sum\limits_{j = {N_{t} - N_{M} + 1}}^{N_{t}}e^{- \frac{{{{\overset{->}{r}}_{o} - {\overset{->}{r}}_{j}}}^{2}}{\lambda_{u}^{2}}}}}} \right)}{Z_{t}}$where

represents a damped mass center of previous locations where emission ofthe emitting source has been detected and λ_(G) represents a scaleassociated with a Gaussian term approximating terms associated withdetection of emission of the emitting source;

represents the locations of the searching system sensor where emissionof the emitting source has not been detected; N_(m) is the number oflocations where measurements have been obtained; λ_(u) represents ascale of a Gaussian term approximating terms associated with absence ofdetection of emission of the emitting source; and Z_(t) is anormalization constant.
 2. The method according to claim 1, wherein thefree energy variation is computed as a function of a first element thatfavors exploration of the searching space and as a function of a secondelement that favors exploitation of previous emission propagationmeasurements obtained along the self-generated path, the first andsecond elements being weighted in relation to each other.
 3. The methodaccording to claim 1, further comprising determining whether the currentsensor location or the next location of the mobile sensor corresponds tothe location of the emitting source.
 4. The method according to claim 3,wherein the obtaining an emission propagation measurement, computing afree energy variation, determining a minimum free energy variationvalue, identifying the next location of the mobile sensor, anddetermining whether the current sensor location or the next location ofthe mobile sensor corresponds to the location of the emitting source arerepeated until the location of the emitting source is located.
 5. Themethod according to claim 1, further comprising comparing the determinedminimum free energy variation value with a predetermined threshold, thenext location of the mobile sensor being identified as the locationassociated with the determined minimum free energy variation or as thecurrent sensor location as a function of the result of the comparison.6. The method according to claim 1, further comprising storing theobtained emission propagation measurement in memory along with thecurrent mobile sensor location.
 7. The method according to claim 1,wherein the obtaining an emission propagation measurement from themobile sensor at the mobile sensor location further comprises obtainingat least an emission propagation measurement from at least a secondmobile sensor, different from the mobile sensor, at the location of theat least a second mobile sensor, the at least a second mobile sensorbeing configured to obtain emission propagation measurements, the atleast a second mobile sensor being mobile along at least a secondself-generated path, the free energy being computed as a function of thestandardized projected probability field of the location of the emittingsource based on previous emission propagation measurements obtainedalong the self-generated path and on at least a previous emissionpropagation measurement obtained along the at least a secondself-generated path.
 8. The method according to claim 1, wherein theemitting source is a source of particles, molecules, or fragments ofmolecules.
 9. The method according to claim 1, wherein the emittingsource is a heat source.
 10. The method according to claim 1, whereinthe emitting source is a source of data, the sensor being responsive todata patterns.
 11. A non-transitory computer-readable storage mediumstoring instructions of a computer program for implementing the methodaccording to claim
 1. 12. An apparatus or a set of apparatus comprisingmeans adapted for carrying out each step of the method according toclaim
 1. 13. The apparatus or the set of apparatus of claim 12,comprising first means embedding the mobile sensor, the first meansbeing mobile in the search space, and second means for computing thefree energy variation, determining a minimum free energy variationvalue, and identifying the next location of the mobile sensor, the firstand second means comprising means for exchanging data with each other.14. A method for locating an emitting source of which a measurement ofemission propagation at a different location from that of the emittingsource can be obtained from that different location, in a searchingspace, lacking space perception, using a mobile sensor configured toobtain emission propagation measurements, the sensor being mobile alonga self-generated path, the method comprising: obtaining an emissionpropagation measurement from the mobile sensor at the mobile sensorlocation; for a plurality of possible next locations of the mobilesensor, computing a free energy variation for moving the sensor from itscurrent location to each of the plurality of possible next locations,the free energy being computed as a function of a standardized projectedprobability field of the location of the emitting source based onprevious emission propagation measurements obtained along theself-generated path without space perception; determining a minimum freeenergy variation value amongst the computed free energy variations;identifying the location associated with the determined minimum freeenergy variation as being the next location of the mobile sensor; andmoving the mobile sensor to the identified next location of the mobilesensor, wherein the standardized projected probability field of thelocation of the emitting source, representing the probability P_(t)^(M)({right arrow over (r)}₀|Θ_(t)) that the emitting source is locatedat location {right arrow over (r)}₀, knowing path Θ_(t), can beexpressed with the following formula:${P_{t}^{M}\left( {\overset{\rightarrow}{r_{0}}❘\Theta_{t}} \right)} = \frac{e^{- \frac{{{{\overset{\rightarrow}{r}}_{0} - \overset{\rightarrow}{r_{G}^{N_{s}}}}}^{2}}{\lambda_{G}^{2}}}\left( {{- \frac{1}{N_{M}}}{\sum\limits_{k = 1}^{N_{s}}{\sum\limits_{j = {N_{t} - {\lbrack\frac{({N_{M} + 1})}{N_{s}}\rbrack}}}^{N_{s}}e^{- \frac{{{\overset{\rightarrow}{r_{0}} - \overset{\rightarrow}{r_{j}^{k}}}}^{2}}{\lambda_{u}^{2}}}}}} \right)}{Z_{t}}$where {right arrow over (r^(N) ^(s) _(G))} represents a damped masscenter of previous locations where emission of the emitting source hasbeen detected by the mobile sensor and the at least a second mobilesensor and λ_(G) represents a scale associated with a Gaussian termapproximating terms associated with detection of emission of theemitting source; {right arrow over (r_(j) ^(k))} represents thelocations of a searching system sensor having index k where emission ofthe emitting source has not been detected; N_(M) is the number oflocations where measurements have been obtained; λ_(u) represents ascale of a Gaussian term approximating terms associated with absence ofdetection of emission of the emitting source; and Z_(t) is anormalization constant.